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In a Hilbert framework, we introduce continuous and discrete dynamical systems which aim at solving inclusions governed by structured monotone operators, whereis the subdifferential of a convex lower semicontinuous function, andis a monotone cocoercive operator. We first consider the extension to this setting of the regularized Newton dynamic with two potentials which was considered in Abbas, Attouch, Svaiter JOTA, 2014. Then, we revisit some related dynamical systems, namely the semigroup of contractions generated by, and the continuous gradient projection dynamic. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems, thereby extending the known results. The time discretization of these dynamics gives various forward–backward splitting methods (some new) for solving structured monotone inclusions involving non-potential terms. The convergence of these algorithms is obtained under classical step size limitation. Perspectives are given in the field of numerical splitting methods for optimization, and multicriteria decision processes. [ABSTRACT FROM AUTHOR]